Find a conformal map $f$ from $\Omega = ${$z \in \mathbb{C} | \operatorname{Re}(z)>0$}$ \setminus [0,1]$ to unit disk $\mathbb{D}$. Also make sure that $f(2) = 0$.
I drew $\Omega$ (see image below, $\Omega$ is the sketched part without the blue line) and from this I want to use a composition of conformal maps to get a conformal map that maps $\Omega$ to the unit disk.
The problem is that I don't know how I can manipulate the split or get rid of it. I know that I can use $$ z \mapsto \frac{az+b}{cz+d}$$ and choose 3 points that I want to map to other points, but also for this I don't know which ones I would use.
Can somebody help me with an idea of how I should get rid of the split? Or is there a common method for conformal maps for domains with splits?